Question
Willow is in her fourth year working for the same company. Her starting salary was $48,000. The next year, her salary was $50,400. The following year, her salary was $52,920. This year, her salary is $55,566. Willow’s boss says her salary will continue to increase by the same percentage each year. (4 points) Write the values given in the scenario as a sequence. Do not use $ in the answers. , , , Question 2 Select the appropriate answer.(1 point) This is an example of sequence. Question 3 Create a function that models this sequence/scenario. (3 points) S(n)= ()n By what percentage is Willow’s salary increasing each year?
Answers
GPT-4o mini
Let's break down the assignment step by step.
### Question 1: Write the values given in the scenario as a sequence.
The salaries over the four years can be written as a sequence:
**Sequence:**
48000, 50400, 52920, 55566
### Question 2: Select the appropriate answer.
This is an example of a **geometric sequence** because Willow's salary increases by a constant percentage each year.
### Question 3: Create a function that models this sequence/scenario.
To model the sequence using a function, we can determine the common ratio (the percentage increase) first. We can calculate the percentage increase year over year.
1. **Calculate the percentage increase between the years:**
- From $48,000 to $50,400:
\[
\text{Percentage Increase} = \left(\frac{50400 - 48000}{48000}\right) \times 100 = \left(\frac{2400}{48000}\right) \times 100 = 5\%
\]
- From $50,400 to $52,920:
\[
\text{Percentage Increase} = \left(\frac{52920 - 50400}{50400}\right) \times 100 = \left(\frac{2520}{50400}\right) \times 100 = 5\%
\]
- From $52,920 to $55,566:
\[
\text{Percentage Increase} = \left(\frac{55566 - 52920}{52920}\right) \times 100 = \left(\frac{2646}{52920}\right) \times 100 = 5\%
\]
The percentage increase each year is consistently **5%**.
Now that we know the percentage increase is 5%, we can create the function.
The formula for a geometric sequence can be given by:
\[
S(n) = a \cdot r^{(n - 1)}
\]
where:
- \( S(n) \) is the salary in the nth year,
- \( a \) is the initial salary ($48,000),
- \( r \) is the common ratio (1 + percentage increase as a decimal; for 5%, it is 1.05),
- \( n \) is the year number.
Thus, the function modeling Willow's salary sequence would be:
**Function:**
\[
S(n) = 48000 \cdot (1.05)^{(n - 1)}
\]
### Summary of Answers:
1. Sequence: **48000, 50400, 52920, 55566**
2. Type of sequence: **Geometric sequence**
3. Function: **S(n) = 48000 * (1.05)^{(n - 1)}**
4. Percentage increase: **5%**
### Question 1: Write the values given in the scenario as a sequence.
The salaries over the four years can be written as a sequence:
**Sequence:**
48000, 50400, 52920, 55566
### Question 2: Select the appropriate answer.
This is an example of a **geometric sequence** because Willow's salary increases by a constant percentage each year.
### Question 3: Create a function that models this sequence/scenario.
To model the sequence using a function, we can determine the common ratio (the percentage increase) first. We can calculate the percentage increase year over year.
1. **Calculate the percentage increase between the years:**
- From $48,000 to $50,400:
\[
\text{Percentage Increase} = \left(\frac{50400 - 48000}{48000}\right) \times 100 = \left(\frac{2400}{48000}\right) \times 100 = 5\%
\]
- From $50,400 to $52,920:
\[
\text{Percentage Increase} = \left(\frac{52920 - 50400}{50400}\right) \times 100 = \left(\frac{2520}{50400}\right) \times 100 = 5\%
\]
- From $52,920 to $55,566:
\[
\text{Percentage Increase} = \left(\frac{55566 - 52920}{52920}\right) \times 100 = \left(\frac{2646}{52920}\right) \times 100 = 5\%
\]
The percentage increase each year is consistently **5%**.
Now that we know the percentage increase is 5%, we can create the function.
The formula for a geometric sequence can be given by:
\[
S(n) = a \cdot r^{(n - 1)}
\]
where:
- \( S(n) \) is the salary in the nth year,
- \( a \) is the initial salary ($48,000),
- \( r \) is the common ratio (1 + percentage increase as a decimal; for 5%, it is 1.05),
- \( n \) is the year number.
Thus, the function modeling Willow's salary sequence would be:
**Function:**
\[
S(n) = 48000 \cdot (1.05)^{(n - 1)}
\]
### Summary of Answers:
1. Sequence: **48000, 50400, 52920, 55566**
2. Type of sequence: **Geometric sequence**
3. Function: **S(n) = 48000 * (1.05)^{(n - 1)}**
4. Percentage increase: **5%**